order	240
isoclinism	isoclinic class of C₂⁴⋊C₅
automorphism	Aut(G) = C₂×AΓL(1,𝔽₁₆) (order 1920) · Inn(G) = C₂⁴⋊C₅ · Out(G) = C₂×Dic₃
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₃×C₂⁴⋊C₅
unambiguous semidirect product	(C₂⁴×C₃)⋊C₅

Splittings

Fitting, socle, Hall	C₂⁴×C₃ ⋊ C₅
derived, Sylow	C₂⁴ ⋊ C₃×C₅

	subgroup	G/H
center, hypercenter	C₃	C₂⁴⋊C₅
commutator, cosocle, hypocenter	C₂⁴	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	C₅
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃

34 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2,4,6,8)(3,5,7,9,10)(11,12,13);(1,3)(8,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 199);